A167560 -id:A167560 - cp's OEIS frontend

A167565 A triangle related to the a(n) formulas for the rows of the ED2 array A167560.

1, 2, 0, 3, 1, 2, 4, 4, 16, 0, 5, 10, 67, 14, 24, 6, 20, 202, 124, 368, 0, 7, 35, 497, 601, 2736, 444, 720, 8, 56, 1064, 2120, 13712, 6464, 16896, 0, 9, 84, 2058, 6096, 53121, 48876, 186732, 25584, 40320, 10, 120, 3684, 15168, 171258, 257640, 1350296

Offset: 1

Johannes W. Meijer, Nov 10 2009

The a(n) formulas given below correspond to the first ten rows of the ED2 array A167560.

The recurrence relations for the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A003148 and A007318.

			Row 1: a(n) = 1.
Row 2: a(n) = 2*n + 0.
Row 3: a(n) = 3*n^2 + 1*n + 2.
Row 4: a(n) = 4*n^3 + 4*n^2 + 16*n + 0.
Row 5: a(n) = 5*n^4 + 10*n^3 + 67*n^2 + 14*n + 24.
Row 6: a(n) = 6*n^5 + 20*n^4 + 202*n^3 + 124*n^2 + 368*n + 0.
Row 7: a(n) = 7*n^6 + 35*n^5 + 497*n^4 + 601*n^3 + 2736*n^2 + 444*n + 720.
Row 8: a(n) = 8*n^7 + 56*n^6 + 1064*n^5 + 2120*n^4 + 13712*n^3 + 6464*n^2 + 16896*n + 0.
Row 9: a(n) = 9*n^8 + 84*n^7 + 2058*n^6 + 6096*n^5 + 53121*n^4 + 48876*n^3 + 186732*n^2 + 25584*n + 40320.
Row 10: a(n) = 10*n^9 + 120*n^8 + 3684*n^7 + 15168*n^6 + 171258*n^5 + 257640*n^4 + 1350296*n^3 + 533472*n^2 + 1297152*n + 0.

A167560 is the ED2 array.
A000012, A005843 (n=>1), 2*A104249 (n=>1), A167561, A167562 and A167563 equal the first sixth rows of the array.
A005359 equals the first right hand triangle column.
A000027, A000292, A167566, A167567 and A168304 equal the first five left hand triangle columns.
A000142 equals the row sums.
Cf. A003148 and A007318.

Comment and links added by Johannes W. Meijer, Nov 23 2009

A167569 The lower left triangle of the ED2 array A167560.

Original entry on oeis.org

1, 2, 4, 6, 16, 32, 24, 80, 192, 384, 120, 480, 1344, 3072, 6144, 720, 3360, 10752, 27648, 61440, 122880, 5040, 26880, 96768, 276480, 675840, 1474560, 2949120, 40320, 241920, 967680, 3041280, 8110080, 19169280, 41287680, 82575360

Offset: 1

Johannes W. Meijer, Nov 10 2009

We discovered that the numbers that appear in the lower left triangle of the ED2 array A167560 (m <= n) behave in a regular way, see the formula below. This rather simple regularity doesn't show up in the upper right triangle of the ED2 array (m > n).

			The first few triangle rows are:
[1]
[2, 4]
[6, 16, 32]
[24, 80, 192, 384]
[120, 480, 1344, 3072, 6144]
[720, 3360, 10752, 27648, 61440, 122880]

G. C. Greubel, Table of n, a(n) for the first 50 rows

A167560 is the ED2 array.
A047053, 2*A034177 and A167570 are the first three right hand triangle columns.
A000142, 4*A001715, 32*A001725, 384* A049388 and 6144* A049398 are the first five left hand triangle columns.
A167571 equals the row sums.

a := proc(n, m): 4^(m-1)*(m-1)!*(n+m-1)!/(2*m-1)! end: seq(seq(a(n, m), m=1..n), n=1..8); # Johannes W. Meijer, revised Nov 23 2012

Flatten[Table[4^(m - 1)*(m - 1)!*(n + m - 1)!/(2*m - 1)!, {n, 1, 50}, {m, n}]] (* G. C. Greubel, Jun 16 2016 *)

a(n,m) = 4^(m-1)*(m-1)!*(n+m-1)!/(2*m-1)!.

A167561 The fourth row of the ED2 array A167560.

Original entry on oeis.org

24, 80, 192, 384, 680, 1104, 1680, 2432, 3384, 4560, 5984, 7680, 9672, 11984, 14640, 17664, 21080, 24912, 29184, 33920, 39144, 44880, 51152, 57984, 65400, 73424, 82080, 91392, 101384, 112080, 123504, 135680, 148632, 162384, 176960

Offset: 1

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A167560.

Table[4*n^3 + 4*n^2 + 16*n, {n, 1, 100}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {24, 80, 192, 384}, 100] (* G. C. Greubel, Jun 15 2016 *)

a(n) = 4*n^3 + 4*n^2 + 16*n.
G.f.: (0*z^3 + 16*z^2 - 16*z + 24)/(1-z)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - G. C. Greubel, Jun 15 2016

A167562 The fifth row of the ED2 array A167560.

Original entry on oeis.org

120, 480, 1344, 3072, 6144, 11160, 18840, 30024, 45672, 66864, 94800, 130800, 176304, 232872, 302184, 386040, 486360, 605184, 744672, 907104, 1094880, 1310520, 1556664, 1836072, 2151624, 2506320, 2903280, 3345744, 3837072

Offset: 1

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Equals the fifth row of the ED2 array A167560.

Table[5*n^4 + 10*n^3 + 67*n^2 + 14*n + 24, {n, 1, 100}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {120, 480, 1344, 3072, 6144}, 100] (* G. C. Greubel, Jun 16 2016 *)

a(n) = 5*n^4 + 10*n^3 + 67*n^2 + 14*n + 24.
G.f.: (24*z^4 - 48*z^3 + 144*z^2 - 120*z + 120)/(1-z)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - G. C. Greubel, Jun 16 2016

A167563 The sixth row of the ED2 array A167560.

Original entry on oeis.org

720, 3360, 10752, 27648, 61440, 122880, 226800, 392832, 646128, 1018080, 1547040, 2279040, 3268512, 4579008, 6283920, 8467200, 11224080, 14661792, 18900288, 24072960, 30327360, 37825920, 46746672, 57283968, 69649200

Offset: 1

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Cf. A167560.

Table[6*n^5 + 20*n^4 + 202*n^3 + 124*n^2 + 368*n, {n,1,100}] (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1},{720, 3360, 10752, 27648, 61440, 122880},100] (* G. C. Greubel, Jun 16 2016 *)

a(n) = 6*n^5 + 20*n^4 + 202*n^3 + 124*n^2 + 368*n.
G.f.: (432*z^4 - 864*z^3 + 1392*z^2 - 960*z + 720)/(1-z)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - G. C. Greubel, Jun 16 2016

A167568 A triangle related to the GF(z) formulas of the rows of the ED2 array A167560.

Original entry on oeis.org

1, 0, 2, 2, -2, 6, 0, 16, -16, 24, 24, -48, 144, -120, 120, 0, 432, -864, 1392, -960, 720, 720, -2160, 8208, -12816, 14448, -8400, 5040, 0, 23040, -69120, 149760, -184320, 161280, -80640, 40320, 40320, -161280, 760320, -1716480, 2684160, -2695680

Offset: 1

Johannes W. Meijer, Nov 10 2009

The GF(z) formulas given below correspond to the first ten rows of the ED2 array A167560. The polynomials in their numerators lead to the triangle given above.

			Row 1: GF(z) = 1/(1-z).
Row 2: GF(z) = 2/(1-z)^2.
Row 3: GF(z) = (2*z^2 - 2*z + 6)/(1-z)^3.
Row 4: GF(z) = (0*z^3 + 16*z^2 - 16*z + 24)/(1-z)^4.
Row 5: GF(z) = (24*z^4 - 48*z^3 + 144*z^2 - 120*z + 120)/(1-z)^5.
Row 6: GF(z) = (432*z^4 - 864*z^3 + 1392*z^2 - 960*z + 720)/(1-z)^6.
Row 7: GF(z) = (720*z^6 - 2160*z^5 + 8208*z^4 - 12816*z^3 + 14448*z^2 - 8400*z + 5040)/(1-z)^7.
Row 8: GF(z) = (0*z^7 + 23040*z^6 - 69120*z^5 + 149760*z^4 - 184320*z^3 + 161280*z^2 - 80640*z + 40320)/(1-z)^8.
Row 9: GF(z) = (40320*z^8 - 161280*z^7 + 760320*z^6 - 1716480*z^5 + 2684160*z^4 - 2695680*z^3 + 1935360*z^2 - 846720*z + 362880)/(1-z)^9.
Row 10: GF(z) = (0*z^9 + 2016000*z^8 - 8064000*z^7 + 22464000*z^6 - 39168000*z^5 + 48360960*z^4 - 40849920*z^3 + 24917760*z^2 - 9676800*z + 3628800)/(1-z)^10.

A167560 is the ED2 array.
A005359 equals the first left hand column.
A000142(n=>1) and 2*A005990 equal the first two right hand columns.
A000142(n=>1) equals the row sums.

A167564 The row sums of the ED2 array A167560 read by antidiagonals.

Original entry on oeis.org

1, 3, 11, 47, 241, 1457, 10223, 81835, 736637, 7366661, 81034027, 972410303, 12641339465, 176978768137, 2654681568551, 42474905237267, 722073389475733, 12997321011978125, 246949099232271587, 4938981984661217191

Offset: 1

Johannes W. Meijer, Nov 10 2009

A167560 is the ED2 array.

A162005 The EG1 triangle.

Original entry on oeis.org

1, 2, 1, 16, 28, 1, 272, 1032, 270, 1, 7936, 52736, 36096, 2456, 1, 353792, 3646208, 4766048, 1035088, 22138, 1, 22368256, 330545664, 704357760, 319830400, 27426960, 199284, 1, 1903757312, 38188155904, 120536980224, 93989648000

Offset: 1

Johannes W. Meijer, Jun 27 2009, Jul 02 2009, Aug 31 2009

We define the EG1 matrix by EG1[2m-1,1] = 2*eta(2m-1) and the recurrence relation EG1[2m-1,n] = EG1[2m-1,n-1] - EG1[2m-3,n-1]/(n-1)^2 for m = -2, -1, 0, 1, 2, .. and n = 2, 3, .., with eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function. For the EG2[2m,n] coefficients see A008955.
The n-th term of the row coefficients EG1[1-2*m,n] for m = 1, 2, .., can be generated with REG1(1-2*m,n) = (-1)^(m+1)*2^(1-m)*ECGP(1-2*m, n)*(1/n)*4^(-n)*(2*n)!/((n-1)!)^2 . For information about the ECGP polynomials see A094665 and the examples below.
We define the o.g.f.s. of the REG1(1-2*m,n) by GFREG1(z,1-2*m) = sum(REG1(1-2*m,n)* z^(n-1), n=1..infinity) for m = 1, 2, .., with GFREG1(z,1-2*m) = (-1)^(m+1)* RG(z,1-2*m)/ (2^(2*m-1)*(1-z)^((2*m+1)/2)). The RG(z,1-2m) polynomials led to the EG1 triangle.
We used the coefficients of the A156919 and A094665 triangles to determine those of the EG1 triangle, see the Maple program. The A156919 triangle gives information about the sums SF(p) = sum(n^(p-1)*4^(-n)*z^(n-1)*(2*n)!/((n-1)!)^2, n=1..infinity) for p= 0, 1, 2, .. .
Contribution from Johannes W. Meijer, Nov 23 2009: (Start)
The EG1 matrix is related to the ED2 array A167560 because sum(EG1(2*m-1,n)*z^(2*m-1), m=1..infinity) = ((2*n-1)!/(4^(n-1)*(n-1)!^2))*int(sinh(y*(2*z))/cosh(y)^(2*n),y=0..infinity).
(End)
Appears to equal triangle A322230 with rows read in reverse order. Triangle A322230 describes the e.g.f. S(x,k) = Integral C(x,k)*D(x,k)^2 dx, such that C(x,k)^2 - S(x,k)^2 = 1, and D(x,k)^2 - k^2*S(x,k)^2 = 1. - Paul D. Hanna, Dec 22 2018
Appears to equal triangle A325220, which has e.g.f. S(x,k) = -i * sn( i * Integral C(x,k) dx, k) such that C(x,k) = cn( i * Integral C(x,k) dx, k), where sn(x,k) and cn(x,k) are Jacobi Elliptic functions. - Paul D. Hanna, Apr 13 2019

			The first few rows of the EG1 triangle are :
[1]
[2, 1]
[16, 28, 1]
[272, 1032, 270, 1]
The first few RG(z,1-2*m) polynomials are:
RG(z,-1) = 1
RG(z,-3) = 2+z
RG(z,-5) = 16+28*z+z^2
RG(z,-7) = 272+1032*z+270*z^2+z^3
The first few GFREG1(z,1-2*m) are:
GFREG1(z,-1) = (1)*(1)/(2*(1-z)^(3/2))
GFREG1(z,-3) = (-1)*(2+z)/(2^3*(1-z)^(5/2))
GFREG1(z,-5) = (1)*(16+28*z+z^2)/( 2^5*(1-z)^(7/2))
GFREG1(z,-7) = (-1)*(272+1032*z+270*z^2+z^3)/(2^7*(1-z)^(9/2))
The first few REG1(1-2*m,n) are:
REG1(-1,n) = (1/1)*(1)*(1/n)*4^(-n)*(2*n)!/(n-1)!^2
REG1(-3,n) = (-1/2)*(n) *(1/n)*4^(-n)*(2*n)!/(n-1)!^2
REG1(-5,n) = (1/4) *(n+3*n^2) *(1/n)*4^(-n)*(2*n)!/(n-1)!^2
REG1(-7,n) = (-1/8)*(4*n+15*n^2+15*n^3) *(1/n)*4^(-n)*(2*n)!/(n-1)!^2
The first few ECGP(1-2*m,n) polynomials are:
ECGP(-1,n) = 1
ECGP(-3,n) = n
ECGP(-5,n) = n+3*n^2
ECGP(-7,n) = 4*n+15*n^2+15*n^3

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.

A079484 equals the row sums.
A000182 (ZAG numbers), A162006 and A162007 equal the first three left hand columns.
A000012, A004004 (2x), A162008, A162009 and A162010 equal the first five right hand columns.
Related to A094665, A083061 and A156919 (DEF triangle).
Cf. A161198 [(1-x)^((-1-2*n)/2)], A008955 (EG2[2m, n])
Cf. A167560 (ED2 array).
Cf. A322230 (reversed rows), A325220.

nmax:=7; mmax := nmax: imax := nmax: T1(0, x) := 1: T1(0, x+1) := 1: for i from 1 to imax do T1(i, x) := expand((2*x+1) * (x+1)*T1(i-1, x+1)-2*x^2*T1(i-1, x)): dx := degree(T1(i, x)): for k from 0 to dx do c(k) := coeff(T1(i, x), x, k) od: T1(i, x+1) := sum(c(j1)*(x+1)^(j1), j1=0..dx): od: for i from 0 to imax do for j from 0 to i do A083061(i, j) := coeff(T1(i, x), x, j) od: od: for n from 0 to nmax do for k from 0 to n do A094665(n+1, k+1) := A083061(n, k) od: od: A094665(0, 0) := 1: for n from 1 to nmax do A094665(n, 0) := 0 od: for m from 1 to mmax do A156919(0, m) := 0 end do: for n from 0 to nmax do A156919(n, 0) := 2^n end do: for n from 1 to nmax do for m from 1 to mmax do A156919(n, m) := (2*m+2)*A156919(n-1, m) + (2*n-2*m+1)*A156919(n-1, m-1) end do end do: for n from 0 to nmax do SF(n) := sum(A156919(n, k1)*z^k1, k1=0..n)/(2^(n+1)*(1-z)^((2*n+3)/2)) od: GFREG1(z, -1) := A156919(0, 0)*A094665 (0, 0) / (2*(1-z)^(3/2)): for m from 2 to nmax do GFREG1(z, 1-2*m) := simplify((-1)^(m+1)*2^(1-m)* sum(A094665(m-1, k2)*SF(k2), k2=1..m-1)) od: for m from 1 to mmax do g(m) := sort((numer ((-1)^(m+1)* GFREG1(z, 1-2*m))), ascending) od: for n from 1 to nmax do for m from 1 to n do a(n, m) := abs(coeff(g(n), z, m-1)) od: od: seq(seq(a(n, m), m=1..n), n=1..nmax);
# Maple program edited by Johannes W. Meijer, Sep 25 2012

A different form of the recurrence relation is EG1[1-2*m,n] = (EG1[3-2*m,n]-EG1[3-2*m,n+1])* (n^2) for m = 2, 3, .., with EG1[ -1,n] = (1/n)*4^(-n)*((2*n)!/(n-1)!^2).

A167546 The ED1 array read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 6, 12, 7, 1, 24, 48, 32, 10, 1, 120, 240, 160, 62, 13, 1, 720, 1440, 960, 384, 102, 16, 1, 5040, 10080, 6720, 2688, 762, 152, 19, 1, 40320, 80640, 53760, 21504, 6144, 1336, 212, 22, 1

Offset: 1

Johannes W. Meijer, Nov 10 2009

The coefficients in the upper right triangle of the ED1 array (m > n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED1 array (m <= n) were found with the recurrence relation, see below. We use for the array rows the letter n (>= 1) and for the array columns the letter m (>= 1).
Our procedure for finding the coefficients in the lower left triangle can be compared with the procedure that De Smit and Lenstra used to fill in the hole in the middle of 'The Print Gallery' by M. C. Escher, see the links. In this lithograph Escher made use of the so-called Droste effect, hence we propose to call this square array of numbers the ED1 array.
For the ED2, ED3 and ED4 arrays see A167560, A167572 and A167584.

			The ED1 array begins with:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1
1, 4, 7, 10, 13, 16, 19, 22, 25, 28
2, 12, 32, 62, 102, 152, 212, 282, 362, 452
6, 48, 160, 384, 762, 1336, 2148, 3240, 4654, 6432
24, 240, 960, 2688, 6144, 12264, 22200, 37320, 59208, 89664
120, 1440, 6720, 21504, 55296, 122880, 245640, 452880, 783144, 1285536

B. de Smit and H.W. Lenstra, The Mathematical Structure of Escher's Print Gallery, Notices of the AMS, Volume 50, Number 4, pp. 446-457, April 2003.
Johannes W. Meijer, The four Escher-Droste arrays, jpg image, Mar 08 2013.
A. Ryabov, P. Chvosta, Tracer dynamics in a single-file system with absorbing boundary, arXiv preprint arXiv:1402.1949 [cond-mat.stat-mech], 2014.

A000012, A016777, 2*A005891, A167547, A167548 and A167549 equal the first sixth rows of the array.
A000142 equals the first column of the array.
A167550 equals the a(n, n+1) diagonal of the array.
A047053 equals the a(n, n) diagonal of the array.
A167558 equals the a(n+1, n) diagonal of the array.
A167551 equals the row sums of the ED1 array read by antidiagonals.
A167552 is a triangle related to the a(n) formulas of rows of the ED1 array.
A167556 is a triangle related to the GF(z) formulas of the rows of the ED1 array.
A167557 is the lower left triangle of the ED1 array.
Cf. A068424 (the (m-1)!/(m-n-1)! factor), A007680 (the (2*n-1)*(n-1)! factor).
Cf. A167560 (ED2 array), A167572 (ED3 array), A167584 (ED4 array).

nmax:=10; mmax:=10; for n from 1 to nmax do for m from 1 to n do a(n,m) := 4^(m-1)*(m-1)!*(n-1+m-1)!/(2*m-2)! od; for m from n+1 to mmax do a(n,m):= (2*n-1)*(n-1)! + sum((-1)^(k-1)*binomial(n-1,k)*a(n,m-k),k=1..n-1) od; od: for n from 1 to nmax do for m from 1 to n do d(n,m):=a(n-m+1,m) od: od: T:=1: for n from 1 to nmax do for m from 1 to n do a(T):= d(n,m): T:=T+1: od: od: seq(a(n),n=1..T-1);

nmax = 10; mmax = 10; For[n = 1, n <= nmax, n++, For[m = 1, m <= n, m++, a[n, m] = 4^(m - 1)*(m - 1)!*((n - 1 + m - 1)!/(2*m - 2)!)]; For[m = n + 1, m <= mmax, m++, a[n, m] = (2*n - 1)*(n - 1)! + Sum[(-1)^(k - 1)*Binomial[n - 1, k]*a[n, m - k], {k, 1, n - 1}]]; ]; For[n = 1, n <= nmax, n++, For[m = 1, m <= n, m++, d[n, m] = a[n - m + 1, m]]; ]; t = 1; For[n = 1, n <= nmax, n++, For[m = 1, m <= n, m++, a[t] = d[n, m]; t = t + 1]]; Table[a[n], {n, 1, t - 1}] (* Jean-François Alcover, Dec 20 2011, translated from Maple *)

a(n,m) = (2*(m-1)!/(m-n-1)!)*Integral_{y>=0} sinh(y*(2*n-1))/cosh(y)^(2*m-1) for m > n.
The (n-1)-differences of the n-th array row lead to the recurrence relation
Sum_{k=0..n-1} (-1)^k*binomial(n-1,k)*a(n,m-k) = (2*n-1)*(n-1)!
which in its turn leads to, see also A167557,
a(n,m) = 4^(m-1)*(m-1)!*(n+m-2)!/(2*m-2)! for m <= n.

A104249 a(n) = (3*n^2 + n + 2)/2.

Original entry on oeis.org

1, 3, 8, 16, 27, 41, 58, 78, 101, 127, 156, 188, 223, 261, 302, 346, 393, 443, 496, 552, 611, 673, 738, 806, 877, 951, 1028, 1108, 1191, 1277, 1366, 1458, 1553, 1651, 1752, 1856, 1963, 2073, 2186, 2302, 2421, 2543, 2668, 2796, 2927, 3061, 3198, 3338, 3481

Offset: 0

Thomas Wieder, Feb 26 2005

Second differences are all 3.
Related to the sequence of odd numbers A005408 since for these numbers the first differences are all 2.
Column 2 of A114202. - Paul Barry, Nov 17 2005
Equals third row of A167560 divided by 2. - Johannes W. Meijer, Nov 12 2009
A242357(a(n)) = n + 1. - Reinhard Zumkeller, May 11 2014
Also, this sequence is related to A011379, for n>0, by A011379(n) = n*a(n) - Sum_{i=0..n-1} a(i). - Bruno Berselli, Jul 08 2015
The number of Hamiltonian nonisomorphic unfoldings in an n-gonal Archimedean antiprism. See sequence A284647. - Rick Mabry, Apr 10 2021

			The sequence of first differences delta_a(n) = a(n+1) - a(n) is 2, 5, 8, 11, 14, 17, 20, 23, 26, ...
The sequence of second differences delta_delta_a(n) = a(n+2) - 2*a(n+1) + a(n) is: 3, 3, 3, 3, 3, 3, 3, ... E.g., 78 - 2*58 + 41 = 3.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Rick Mabry, Fibonacci Numbers, Integer Compositions, and Nets of Antiprisms, The American Mathematical Monthly, Vol. 126 (2019), no. 9, pp. 786-801.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001399, A002597, A005408, A011379, A016777, A143689.

Counts special cases of A284647.

a104249 n = n*(3*n+1) `div` 2 + 1 -- Reinhard Zumkeller, May 11 2014

[(3*n^2+n+2)/2: n in [0..50]]; // Vincenzo Librandi, May 09 2011

a := proc (n) local i, u; option remember; u[0] := 1; u[1] := 3; u[2] := 8; for i from 3 to n do u[i] := -(4*u[i-3]-8*u[i-2]-2*u[i-1]+(-2*u[i-3]+2*u[i-2]-u[i-1])*i)/i end do; [seq(u[i],i = 0 .. n)] end proc;

A104249[n_] := (3*n^2 + n + 2)/2; Table[A104249[n], {n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)
LinearRecurrence[{3,-3,1},{1,3,8},70] (* Harvey P. Dale, Jul 21 2023 *)

a(n)=n*(3*n+1)/2+1 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1 + 2*x^2)/(1 - x)^3.
Recurrence: (n+3)*u(n+3) + (-5-n)*u(n+2)*(-2+2*n)*u(n+1) + (-2-2*n)*u(n) = 0 for n >= 0 with u(0) = 1, u(1) = 3, and u(2) = 8.
From Paul Barry, Nov 17 2005: (Start)
a(0) = 1, a(n) = a(n-1) + 3*n - 1 for n > 0;
a(n) = Sum_{k=0..n} C(n, k)*C(2, k)*J(k+1), where J(n) = A001045(n). (End)
Binomial transform of [1, 2, 3, 0, 0, 0, ...]. - Gary W. Adamson, Apr 23 2008
E.g.f.: exp(x)*(2 + 4*x + 3*x^2)/2. - Stefano Spezia, Apr 10 2021

cp's OEIS Frontend

A167565 A triangle related to the a(n) formulas for the rows of the ED2 array A167560.

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A167569 The lower left triangle of the ED2 array A167560.

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A167561 The fourth row of the ED2 array A167560.

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A167562 The fifth row of the ED2 array A167560.

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A167563 The sixth row of the ED2 array A167560.

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A167568 A triangle related to the GF(z) formulas of the rows of the ED2 array A167560.

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A167564 The row sums of the ED2 array A167560 read by antidiagonals.

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A162005 The EG1 triangle.

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A167546 The ED1 array read by antidiagonals.

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