A064226 -id:A064226 - cp's OEIS frontend

A081269 Duplicate of A064226.

3, 14, 34, 63, 101, 148, 204, 269, 343, 426, 518, 619, 729, 848, 976, 1113, 1259, 1414

Offset: 0

dead

A064225 a(n) = (9n^2 + 5n + 2)/2.

1, 8, 24, 49, 83, 126, 178, 239, 309, 388, 476, 573, 679, 794, 918, 1051, 1193, 1344, 1504, 1673, 1851, 2038, 2234, 2439, 2653, 2876, 3108, 3349, 3599, 3858, 4126, 4403, 4689, 4984, 5288, 5601, 5923, 6254, 6594, 6943, 7301, 7668, 8044, 8429, 8823, 9226

Offset: 0

N. J. A. Sloane, Sep 22 2001

Diagonal of triangular spiral in A051682. - Michael Somos, Jul 22 2006
Ehrhart polynomial of closed quadrilateral with vertices (0,2),(2,3),(3,1),(2,0). - Michael Somos, Jul 22 2006
In the natural number array A000027 this sequence is the first knight moves diagonal (A081267 is the second, A001844 is the main diagonal). It can be used to define this diagonal for any array: A007318(A064225-1)=A005809 (Subtraction by 1 because A007318 is defined with offset 0.) - Tilman Piesk, Mar 24 2012
Or positions of pentagonal numbers, such that p(a(n)) = p(a(n)-1) + p(3*n+1), where p=A000326. - Vladimir Shevelev, Jan 21 2014

			Illustration of initial terms:
.
.                                    o
.                                 o o
.                      o       o o o o
.                   o o     o o o o o
.           o    o o o o     o o o o o
.        o o      o o o     o o o o o
.   o     o o    o o o o     o o o o o
.        o o      o o o     o o o o o
.           o    o o o o     o o o o o
.                   o o     o o o o o
.                      o       o o o o
.                                 o o
.                                    o
.
.   1     8        24           49
- _Aaron David Fairbanks_, Feb 23 2025

Harry J. Smith, Table of n, a(n) for n = 0..1000
National Security Agency, Intrigued? (advertisement), Notices of the Amer. Math. Soc., vol. 49 (2002), p. 216.
J. A. Siehler, Selections without adjacency on a rectangular grid, arXiv:1409.3869, Table 3, k=2 (different offset)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000027, A000326, A001844, A005809, A007318, A051682, A064226, A081267, A235332.

Column w=2 of A371967.

Table[(9n^2+5n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,8,24},51] (* Harvey P. Dale, Sep 13 2011 *)

{a(n) = 1 + n * (9*n + 5) / 2}; /* Michael Somos, Jul 22 2006 */

(define (A064225 n) (/ (+ (* 9 n n) (* 5 n) 2) 2))

a(n) = 9*n+a(n-1)-2, with n>0, a(0) = 1. - Vincenzo Librandi, Aug 07 2010
a(0)=1, a(1)=8, a(2)=24, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Sep 13 2011
G.f.: (1+5*x+3*x^2)/(1-x)^3. - Colin Barker, Feb 23 2012
A064226(n) = a(-1-n). - Michael Somos, Jul 22 2006 (While the sequence itself is only one-way infinite, this identity works, as the defining formula (in the Name-field) produces integers also for the negative values of n, -1, -2, -3, etc.) - Antti Karttunen, Mar 24 2012
E.g.f.: exp(x)*(2 + 14*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022

A140064 a(n) = (9n^2 - 5n + 2)/2.

Original entry on oeis.org

1, 3, 14, 34, 63, 101, 148, 204, 269, 343, 426, 518, 619, 729, 848, 976, 1113, 1259, 1414, 1578, 1751, 1933, 2124, 2324, 2533, 2751, 2978, 3214, 3459, 3713, 3976, 4248, 4529, 4819, 5118, 5426, 5743, 6069, 6404, 6748, 7101, 7463, 7834, 8214, 8603, 9001, 9408, 9824, 10249, 10683, 11126, 11578, 12039, 12509, 12988, 13476, 13973

Offset: 0

Gary W. Adamson, May 03 2008

Originally this entry was defined by a(n) = (9*n^2 - 23*n + 16)/2 and had offset 1. The current, simpler definition seems preferable, since it matches the following geometrical applications. This change will also require several changes to the rest of the entry. - N. J. A. Sloane, Jun 26 2025
The letter Wu, ᗐ, is like a V but with three arms instead of two. Wu is included in the Unified Canadian Aboriginal Syllabics section of Unicode. The Unicode symbol for Wu is 0x2a5b. Wu is also called a "Boolean OR with middle stem", and is also the alchemical symbol Dissolve-2.
The formal definition is that a long-legged Wu is a pencil of three semi-infinite lines originating from a point (the "tip"). The angles between the three lines are arbitrary.
Theorem 1 (Edward Xiong, Jonathan Pei, and David Cutler, Jun 24 2025): a(n) is the maximum number of regions in the plane that can be formed from n copies of a long-legged Wu.
Theorem 2: a(n) is also the maximum number of regions in the plane that can be formed from n copies of a long-legged letter A.
For proofs of Theorems 1 and 2 see "Cutting a pancake with an exotic knife".
For analogous sequences for long-legged letters V and Z see A130883 and A117625.

David O. H. Cutler and Neil J. A. Sloane, Cutting a pancake with an exotic knife, Paper in preparation, Sep 05 2025

N. J. A. Sloane, Table of n, a(n) for n = 0..5000 [First 1000 terms from G. C. Greubel]
N. J. A. Sloane, The long-legged letters A, I, V, X, and Z. The long-legged A is a long-legged V with a crossbar. The distances from the tip of the A to the end-points of the crossbar need not be equal.
N. J. A. Sloane, 14 regions using 2 copies of the Wu graph
N. J. A. Sloane, 34 regions using 3 copies of the Wu graph
N. J. A. Sloane, Transforming a Hatpin graph to a Wu graph.
N. J. A. Sloane, 14 regions using 2 long-legged A's [Crossbars are shown in red]
N. J. A. Sloane, 34 regions using 3 long-legged A's [Crossbars are shown in red]
N. J. A. Sloane, Transforming a long-legged A_n graph to a Wu_n graph, and vice-versa
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A007318, A064226, A130883, A117625.

A row of the array in A386478.

[ n eq 1 select 1 else Self(n-1)+9*n-16: n in [1..50] ];

seq((16-23*n+9*n^2)*1/2,n=1..40); # Emeric Deutsch, May 07 2008

Table[(9n^2-23n+16)/2,{n,40}] (* or *) LinearRecurrence[{3,-3,1},{1,3,14},40] (* Harvey P. Dale, Oct 01 2011 *)

x='x+O('x^50); Vec(x*(1+8*x^2)/(1-x)^3) \\ G. C. Greubel, Feb 18 2017

Binomial transform of [1, 2, 9, 0, 0, 0, ...].
a(n) = A000217(n) + 8*A000217(n-2). - R. J. Mathar, May 06 2008
O.g.f.: x*(1+8*x^2)/(1-x)^3. - R. J. Mathar, May 06 2008
a(n) = A064226(n-2), n>1. - R. J. Mathar, Jul 31 2008
a(n) = a(n-1) + 9*n - 16, a(1)=1. - Vincenzo Librandi, Nov 24 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=3, a(3)=14. - Harvey P. Dale, Oct 01 2011
E.g.f.: exp(x)*(16 - 14*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022

More terms from R. J. Mathar and Emeric Deutsch, May 06 2008

Edited by N. J. A. Sloane, Jun 21 2025 and Jun 26 2025

A235332 a(n) = n(9n + 25)/2 + 6.

Original entry on oeis.org

6, 23, 49, 84, 128, 181, 243, 314, 394, 483, 581, 688, 804, 929, 1063, 1206, 1358, 1519, 1689, 1868, 2056, 2253, 2459, 2674, 2898, 3131, 3373, 3624, 3884, 4153, 4431, 4718, 5014, 5319, 5633, 5956, 6288, 6629, 6979, 7338, 7706, 8083, 8469, 8864, 9268, 9681, 10103

Offset: 0

Bruno Berselli, Jan 22 2014

This is the case d=6 of n*(9*n + 4*d + 1)/2 + d. Other similar sequences are:
d=0, A022267;
d=1, A064225;
d=2, A062123;
d=3, A064226;
d=4, A022266 (with initial 0);
d=5, A178977.
First bisection of A235537.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A017257 (first differences), A022266, A022267, A062123, A064225, A064226, A081267, A178977, A235537.

Magma
```
[n*(9*n+25)/2+6: n in [0..50]];
```

Table[n (9 n + 25)/2 + 6, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{6,23,49},50] (* Harvey P. Dale, Feb 12 2022 *)

a(n)=n*(9*n+25)/2+6 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (6 + 5*x - 2*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
2*a(n) - a(n+1) + 12 = A081267(n).
E.g.f.: exp(x)*(12 + 34*x + 9*x^2)/2. - Elmo R. Oliveira, Nov 13 2024

A081268 Diagonal of triangular spiral in A051682.

Original entry on oeis.org

1, 12, 32, 61, 99, 146, 202, 267, 341, 424, 516, 617, 727, 846, 974, 1111, 1257, 1412, 1576, 1749, 1931, 2122, 2322, 2531, 2749, 2976, 3212, 3457, 3711, 3974, 4246, 4527, 4817, 5116, 5424, 5741, 6067, 6402, 6746, 7099, 7461, 7832, 8212, 8601, 8999, 9406

Offset: 0

Paul Barry, Mar 15 2003

			a(1) = 9*1 +  1 + 2 = 12.
a(2) = 9*2 + 12 + 2 = 32.
a(3) = 9*3 + 32 + 2 = 61.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A064226, A081267.

a(n)=(9*n^2+13*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = C(n,0) + 11*C(n,1) + 9*C(n,2); binomial transform of (1, 11, 9, 0, 0, 0, ...).
a(n) = (9*n^2 + 13*n + 2)/2.
G.f.: (1 + 9*x - x^2)/(1-x)^3.
a(n) = 9*n + a(n-1) + 2 (with a(0)=1). - Vincenzo Librandi, Aug 08 2010
From Elmo R. Oliveira, Nov 16 2024: (Start)
E.g.f.: exp(x)*(9*x^2 + 22*x + 2)/2.
a(n) = A064226(n) - 2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A081270 Diagonal of triangular spiral in A051682.

Original entry on oeis.org

3, 16, 38, 69, 109, 158, 216, 283, 359, 444, 538, 641, 753, 874, 1004, 1143, 1291, 1448, 1614, 1789, 1973, 2166, 2368, 2579, 2799, 3028, 3266, 3513, 3769, 4034, 4308, 4591, 4883, 5184, 5494, 5813, 6141, 6478, 6824, 7179, 7543, 7916, 8298, 8689, 9089, 9498, 9916

Offset: 0

Paul Barry, Mar 15 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hacène Belbachir, Toufik Djellal, Jean-Gabriel Luque, On the self-convolution of generalized Fibonacci numbers, arXiv:1703.00323 [math.CO], 2017.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A064226, A081268.

[(9*n^2+17*n+6)/2: n in [0..50]]; // Vincenzo Librandi, Jul 08 2012

CoefficientList[Series[(3+7x-x^2)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)

a(n)=(9*n^2+17*n+6)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A064226(n) + 2*n.
a(n) = 3*binomial(n,0) + 13*binomial(n,1) + 9*binomial(n,2); binomial transform of (3, 13, 9, 0, 0, 0, ...).
a(n) = (9*n^2 + 17*n + 6)/2.
G.f.: (3 + 7*x - x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 08 2012
E.g.f.: exp(x)*(6 + 26*x + 9*x^2)/2. - Elmo R. Oliveira, Nov 13 2024

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

Original entry on oeis.org

2, 4, 12, 18, 31, 41, 59, 73, 96, 114, 142, 164, 197, 223, 261, 291, 334, 368, 416, 454, 507, 549, 607, 653, 716, 766, 834, 888, 961, 1019, 1097, 1159, 1242, 1308, 1396, 1466, 1559, 1633, 1731, 1809, 1912, 1994, 2102, 2188, 2301, 2391, 2509, 2603, 2726, 2824, 2952

Offset: 0

Bruno Berselli, Oct 25 2011

For an origin of this sequence, see the triangular spiral illustrated in the Links section.

First bisection gives A117625 (without the initial term).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A152832 (by Superseeker).

Cf. sequences related to the triangular spiral: A022266, A022267, A027468, A038764, A045946, A051682, A062708, A062725, A062728, A062741, A064225, A064226, A081266-A081268, A081270-A081272, A081275 [incomplete list].

[(6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1: n in [0..50]];

LinearRecurrence[{1,2,-2,-1,1},{2,4,12,18,31},60] (* Harvey P. Dale, Jun 15 2022 *)

for(n=0, 50, print1((6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1", "));

G.f.: (2+2*x+4*x^2+2*x^3-x^4)/((1+x)^2*(1-x)^3).
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
a(n)-a(-n-1) = A168329(n+1).
a(n)+a(n-1) = A102214(n).
a(2n)-a(2n-1) = A016885(n).
a(2n+1)-a(2n) = A016825(n).

cp's OEIS Frontend

A081269 Duplicate of A064226.

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A064225 a(n) = (9*n^2 + 5*n + 2)/2.

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A140064 a(n) = (9*n^2 - 5*n + 2)/2.

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A235332 a(n) = n*(9*n + 25)/2 + 6.

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A081268 Diagonal of triangular spiral in A051682.

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A081270 Diagonal of triangular spiral in A051682.

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A198392 a(n) = (6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16 + 1.

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A064225 a(n) = (9n^2 + 5n + 2)/2.

A140064 a(n) = (9n^2 - 5n + 2)/2.

A235332 a(n) = n(9n + 25)/2 + 6.

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.