cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A162012 The sequence of the absolute values of the a(n-2) coefficients of A162011.

Original entry on oeis.org

0, 19, 663, 6501, 36729, 149842, 491274, 1375206, 3413982, 7710813, 16133689, 31690659, 59028879, 105082068, 179893252, 297641916, 477906924, 747198807, 1140797259, 1704931921, 2499346773, 3600290694, 5103978990, 7130572930
Offset: 1

Views

Author

Johannes W. Meijer, Jun 27 2009

Keywords

Crossrefs

Equals the absolute values of the coefficients that precede the a(n-2) factors of the recurrence relations RR(n) of A162011.
Cf. A006324 [a(n-1)] and A162013 [a(n-3)].

Programs

  • Maple
    nmax:=26; for n from 1 to nmax do RR(n) := expand(product((1-(2*k-1)^2*z)^(n-k+1),k=1..n),z) od: T:=1: for n from 1 to nmax do a(T):=coeff(RR(n),z,2): T:=T+1 od: seq(a(k),k=1..T-1);

Formula

a(n) = (20*n^8+80*n^7+4*n^6-268*n^5-155*n^4+230*n^3+131*n^2-42*n)/360
Recurrence relation sum((-1)^k*binomial(9,k)*a(n-k), k= 0 .. 9) = 0
GF(z) = z*(19+492*z+1218*z^2+492*z^3+19*z^4)/(1-z)^9

A162013 The sequence of the absolute values of the a(n-3) coefficients of A162011.

Original entry on oeis.org

0, 9, 3748, 163160, 2549775, 22768402, 141820764, 685234196, 2738273230, 9438613635, 28894483904, 80240970524, 205377597269, 490460693060, 1103418293480, 2356809738456, 4809498575164, 9426116131517, 17820475867500
Offset: 1

Views

Author

Johannes W. Meijer, Jun 27 2009

Keywords

Crossrefs

Equals the absolute values of the coefficients that precede the a(n-3) factors of the recurrence relations RR(n) of A162011.
Cf. A006324 [a(n-1)] and A162012 [a(n-2)].

Programs

  • Maple
    nmax:=21; for n from 1 to nmax do RR(n) := expand(product((1-(2*k-1)^2*z)^(n-k+1),k=1..n),z) od: T:=1: for n from 1 to nmax do a(T):=coeff(-RR(n),z,3): T:=T+1 od: seq(a(k),k=1..T-1);

Formula

a(n) = (280*n^12+1680*n^11-252*n^10-16660*n^9-13758*n^8+63408*n^7+68705*n^6-104265*n^5-111657*n^4+66997*n^3+56682*n^2-11160*n)/45360
Recurrence relation sum((-1)^k*binomial(13,k)*a(n-k), k= 0..13) = 0
GF(z) = z*(9+3631*z+115138*z^2+718465*z^3+1282314*z^4+718465*z^5+115138*z^6+ 3631*z^7+ 9*z^8)/(1-z)^13

A006324 a(n) = n*(n + 1)*(2*n^2 + 2*n - 1)/6.

Original entry on oeis.org

1, 11, 46, 130, 295, 581, 1036, 1716, 2685, 4015, 5786, 8086, 11011, 14665, 19160, 24616, 31161, 38931, 48070, 58730, 71071, 85261, 101476, 119900, 140725, 164151, 190386, 219646, 252155, 288145, 327856, 371536, 419441, 471835, 528990, 591186
Offset: 1

Views

Author

Albert Rich (Albert_Rich(AT)msn.com), Jun 14 1998

Keywords

Comments

4-dimensional analog of centered polygonal numbers.
Partial sums of A000447. - Zak Seidov, May 19 2006
From Johannes W. Meijer, Jun 27 2009: (Start)
Equals the absolute values of the coefficients that precede the a(n-1) factors of the recurrence relations RR(n) of A162011.
This sequence enabled the analysis of A162012 and A162013. (End)
Equals the number of integer quadruples (x,y,z,w) such that min(x,y) < min(z,w), max(x,y) < max(z,w), and 0 <= x,y,z,w <= n. - Andrew Woods, Apr 21 2014
For n>3 a(n)=twice the area of an irregular quadrilateral with vertices at the points (C(n,4),C(n+1,4)), (C(n+1,4),C(n+2,4)), (C(n+2,4),C(n+3,4)), and (C(n+3,4),C(n+4,4)). - J. M. Bergot, Jun 14 2014

Links

Crossrefs

Cf. A000447, A000539, A000217, A002415, A053134.
Cf. A162011, A162012, a(n-2), and A162013, a(n-3). - Johannes W. Meijer, Jun 27 2009

Programs

Formula

a(n) = 8*C(n + 2, 4) + C(n + 1, 2).
a(n) = (Sum_{k=1..n} k^5) / (Sum_{k=1..n} k) = A000539(n) / A000217(n). - Alexander Adamchuk, Apr 12 2006
From Johannes W. Meijer, Jun 27 2009: (Start)
Recurrence relation 0 = Sum_{k=0..5} (-1)^k*binomial(5,k)*a(n-k).
G.f.: (1+6*z+z^2)/(1-z)^5. (End)
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Wesley Ivan Hurt, May 02 2021
Sum_{n>=1} 1/a(n) = 6 + 2*sqrt(3)*Pi*tan(sqrt(3)*Pi/2). - Amiram Eldar, Aug 23 2022
a(n) = A053134(n-1) - 4*A002415(n). - Yasser Arath Chavez Reyes, Feb 12 2024

Extensions

Simpler definition from Alexander Adamchuk, Apr 12 2006
More terms from Zak Seidov

A004004 a(n) = (3^(2*n+1) - 8*n - 3)/16.

Original entry on oeis.org

0, 1, 14, 135, 1228, 11069, 99642, 896803, 8071256, 72641337, 653772070, 5883948671, 52955538084, 476599842805, 4289398585298, 38604587267739, 347441285409712, 3126971568687473, 28142744118187326, 253284697063686007, 2279562273573174140, 20516060462158567341
Offset: 0

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

The o.g.f. of this sequence enabled the analysis of A162008, A162009 and A162010. - Johannes W. Meijer, Jun 27 2009

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

From Johannes W. Meijer, Jun 27 2009: (Start)
Equals the second right hand column of triangle A162005 divided by 2.
Cf. A162008, A162009, A162010, A162011 and A162014 [2*(1+3*z)].
(End)

Programs

  • Mathematica
    LinearRecurrence[{11, -19, 9}, {0, 1, 14}, 100] (* G. C. Greubel, Jul 06 2016 *)
Table[(3^(2 n + 1) - 8 n - 3)/16, {n, 0, 24}] (* Michael De Vlieger, Jul 08 2016 *)

Formula

G.f.: -x*(1+3*x)/(9*x-1)/(x-1)^2. - Simon Plouffe in his 1992 dissertation.
a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3). - Johannes W. Meijer, Jun 27 2009
a(n) = a(n-1) + (3^(2*n-1) - 1)/2. - Lechoslaw Ratajczak, Jul 06 2016
E.g.f.: (-3 - 8*x + 3*exp(8*x))*exp(x)/16. - Ilya Gutkovskiy, Jul 07 2016

A162008 Third right hand column of the EG1 triangle A162005.

Original entry on oeis.org

16, 1032, 36096, 1035088, 27426960, 702812568, 17753262208, 445736371872, 11162877175440, 279268061007400, 6983654996144256, 174610650719469552, 4365455001524490256, 109138210900706764728, 2728473030627812279040
Offset: 1

Views

Author

Johannes W. Meijer, Jun 27 2009

Keywords

Links

Crossrefs

Third right hand column of the EG1 triangle A162005.
Other right hand columns are A004004 (2x), A162009 and A162010.
Cf. A162011 and A162014.

Formula

a(n) = (64*n^2+112*n+30-432*n*9^n-405*9^n+375*25^n)/128.
a(n) = 46*a(n-1)-663*a(n-2)+3748*a(n-3)-7711*a(n-4)+6606*a(n-5)-2025*a(n-6).
G.f.: (16+296*z-768*z^2-1080*z^3)/((1-z)^3*(1-9*z)^2*(1-25*z)).

A162009 Fourth right hand column of the EG1 triangle A162005.

Original entry on oeis.org

272, 52736, 4766048, 319830400, 18598875760, 1002968825344, 51882638754240, 2621627565515520, 130715075544000720, 6468157990602644480, 318685706549526508832, 15663443488266952501376, 768809642314801857986608
Offset: 1

Views

Author

Johannes W. Meijer, Jun 27 2009

Keywords

Crossrefs

Fourth right hand column of the EG1 triangle A162005.
Other right hand columns are A004004 (2x), A162008 and A162010.
Cf. A162011 and A162014.

Formula

a(n) = (252105*49^n-525+76545*9^n-328125*25^n-225000*25^n*n-2320*n+134136*9^n*n-2112*n^2+46656*9^n*n^2-512*n^3)/3072
a(n) = 130*a(n-1)-6501*a(n-2)+163160*a(n-3)-2236466*a(n-4)+17123340*a(n-5)-71497186*a(n-6)+154127320*a(n-7)-174334221*a(n-8)+98986050*a(n-9)-22325625*a(n-10)
GF(z) = (272+17376*z-321360*z^2-1298624*z^3+8914800*z^4-11262240*z^5-10206000*z^6)/((1-z)^4*(1-9*z)^3*(1-25*z)^2*(1-49*z))

A162010 Fifth right hand column of the EG1 triangle A162005.

Original entry on oeis.org

7936, 3646208, 704357760, 93989648000, 10324483102720, 1013356176688128, 92857038223998720, 8148225153293502720, 695389790665420312320, 58282750219059501633280, 4827428305286309709508736
Offset: 1

Views

Author

Johannes W. Meijer, Jun 27 2009

Keywords

Crossrefs

Fifth right hand column of the EG1 triangle A162005.
Other right hand columns are A004004 (2x), A162008 and A162009.
Cf. A162011 and A162014.

Formula

a(n) = (13230+75008*n^2+65184*n-778248135*49^n+30720*n^3+ 502211745*81^n+ 295312500*25^n-26034048*9^n*n^2-43600032*9^n*n-395300640*49^n*n-19289340*9^n+ 4096*n^4+ 352500000*25^n*n+90000000*25^n*n^2-4478976*9^n*n^3)/98304
a(n)= 295*a(n-1)-36729*a(n-2)+2549775*a(n-3)-109746165*a(n-4)+3080128275*a(n-5)-57713313405*a(n-6)+727045264875*a(n-7)-6122436806115*a(n-8)+33837597147925*a(n-9)-119061300168619*a(n-10)+257794693911405*a(n-11)-339251103039591*a(n-12)+264193039731825*a(n-13)-112000136889375*a(n-14)+19937341265625*a(n-15)
GF(z) = (7936+1305088*z-79792256*z^2-109331968*z^3+41828672000*z^4-460917924352*z^5+238697445120*z^6+5066784271872*z^7-14723693948160*z^8+12172737024000*z^9+8101522800000*z^10)/(1-z)^5/(1-9*z)^4/(1-25*z)^3/(1-49*z)^2/(1-81*z)
Showing 1-7 of 7 results.

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